| L(s) = 1 | − 2-s + 4-s + 4·5-s − 8-s − 3·9-s − 4·10-s − 11-s − 2·13-s + 16-s + 4·17-s + 3·18-s + 6·19-s + 4·20-s + 22-s + 4·23-s + 11·25-s + 2·26-s − 2·29-s + 2·31-s − 32-s − 4·34-s − 3·36-s + 10·37-s − 6·38-s − 4·40-s − 4·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 9-s − 1.26·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.970·17-s + 0.707·18-s + 1.37·19-s + 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s + 0.392·26-s − 0.371·29-s + 0.359·31-s − 0.176·32-s − 0.685·34-s − 1/2·36-s + 1.64·37-s − 0.973·38-s − 0.632·40-s − 0.624·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.550864759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550864759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.835667570086782401538600627509, −9.244985807430746847903326312339, −8.407776312440049587363031299780, −7.43367063727000598425099414193, −6.49670623220275494789529055463, −5.56906145885328447729868384981, −5.19457542288727538942173189487, −3.12395282147585925662067905416, −2.40161210210218469860001168400, −1.12629880107653357251912906599,
1.12629880107653357251912906599, 2.40161210210218469860001168400, 3.12395282147585925662067905416, 5.19457542288727538942173189487, 5.56906145885328447729868384981, 6.49670623220275494789529055463, 7.43367063727000598425099414193, 8.407776312440049587363031299780, 9.244985807430746847903326312339, 9.835667570086782401538600627509
